Montgomery's conjecture for primes in arithmetic progressions

Let $a<q$ be integers with $\gcd(a,q)=1$, and let $q\leq x$. Write $\psi(x,q,a)$ for the weighted prime-counting function in the progression $a\pmod q$, $\pi(x,q,a)$ for the corresponding prime-counting function, and $\varphi(q)$ for Euler's totient. Montgomery's conjecture. For every $\varepsilon>0$,

$$\psi(x,q,a)=\frac{x}{\varphi(q)}+O\left(\frac{x^{1/2+\varepsilon}}{q^{1/2}}\right),$$

and

$$\pi(x,q,a)=\frac{x}{\varphi(q)\log x}+O\left(\frac{x^{1/2+\varepsilon}}{q^{1/2}}\right).$$

This conjecture gives square-root-scale error terms uniformly for moduli up to $x$ and is presented as a strengthening of the conditional estimates obtained from the Riemann hypothesis. The source says that it does not seem to have a similar elementary proof derived directly from the Riemann hypothesis.